Incomplete Kloosterman sums and their applications (Q2759285)

Incomplete Kloosterman sums are of the form \(\sum _{n\leq x, \;(n,m)=1} e((an^*+bn)/m))\), where \(nn^*\equiv 1 \pmod{m}\) and \(x<m\). A nontrivial estimate for these sums has been obtained by A. A. Karatsuba for \(x \geq x^{\varepsilon}\), for any fixed \(\varepsilon >0\). This implies an asymptotic formula for the sum of the fractional parts of \((an^*+bn)/m\) over a similar range. The author extends the admissible range for \(x\) in the latter result to \(x \geq \exp (c\log m/\log \log m)\), where \(c>\log 2\). The main idea of the complicated argument is to consider incomplete Kloosterman sums over ``good'' integers \(n\), estimating the remaining contribution trivially. For certain moduli (in particular, for the primes), the lower bound for \(x\) can be reduced further to \(\exp ((\log m)^{4/5}(\log \log m)^{73/5})\).